Quantum versus Classical Mechanics and Integrability Problems

towards a unification of approaches and tools
1. Auflage 2019
ISBN: 978-3-030-18378-3
Verlag: Springer

Buch, Englisch, 460 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 823 g

This accessible monograph introduces physicists to the general relation between classical and quantum mechanics based on the mathematical idea of deformation quantization and describes an original approach to the theory of quantum integrable systems developed by the author.The first goal of the book is to develop of a common, coordinate free formulation of classical and quantum Hamiltonian mechanics, framed in common mathematical language.In particular, a coordinate free model of quantum Hamiltonian systems in Riemannian spaces is formulated, based on the mathematical idea of deformation quantization, as a complete physical theory with an appropriate mathematical accuracy.The second goal is to develop of a theory which allows for a deeper understanding of classical and quantum integrability. For this reason the modern separability theory on both classical and quantum level is presented. In particular, the book presents a modern geometric separability theory, based on bi-Poissonian and bi-presymplectic representations of finite dimensional Liouville integrable systems and their admissible separable quantizations.The book contains also a generalized theory of classical Stäckel transforms  and the discussion of the concept of quantum trajectories.In order to make the text consistent and self-contained, the book starts with a compact overview of mathematical tools necessary for understanding the remaining part of the book. However, because the book is dedicated mainly to physicists, despite its mathematical nature, it refrains from highlighting definitions, theorems or lemmas.Nevertheless, all statements presented are either proved or the reader is referred to the literature where the proof is available.

Zielgruppe

Research

Autoren/Hrsg.

Blaszak, Maciej

Fachgebiete

• Naturwissenschaften Physik Quantenphysik
• Naturwissenschaften Physik Mechanik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

Introduction.- Basic Mathematical Tools.- Classical Hamiltonian Mechanics.- Classical Integrable and Separable Hamiltonian Systems.- Classical Separability Theory.- Deformation Theory of Classical Poisson Algebras.- Quantum Hamiltonian Mechanics on Symplectic Manifolds.- Position Representation of Quantum Mechanics over Riemannian Configuration Space.- References.- Index